Planar K-Path in Subexponential Time and Polynomial Space

Planar k-Path in Subexponential Time and Polynomial Space

Daniel Lokshtanov1, Matthias Mnich2, and Saket Saurabh3

1 University of California, San Diego, USA, [email protected] 2 International Computer Science Institute, Berkeley, USA, [email protected] 3 The Institute of Mathematical Sciences, Chennai, India, [email protected]

Abstract. In the k-Path problem we are given an n-vertex graph G together with an integer k and asked whether G contains a path of length k as a subgraph. We give the ﬁrst subexponential time, polynomial space parameterized algorithm for k-Path on planar graphs, and more generally, on H-minor-free graphs. The running time of our algorithm is √ 2 O(2O( k log k)nO(1)).

1 Introduction

In the k-Path problem we are given a n-vertex graph G and integer k and asked whether G contains a path of length k as a subgraph. The problem is a gen- eralization of the classical Hamiltonian Path problem, which is known to be NP-complete [16] even when restricted to planar graphs [15]. On the other hand k-Path is known to admit a subexponential time parameterized algorithm when the input is restricted to planar, or more generally H-minor free graphs [4]. For the case of k-Path a subexponential time parameterized algorithm means an algorithm with running time 2o(k)nO(1). More generally, in parameterized complexity problem instances come equipped with a parameter k and a problem is said to be ﬁxed parameter tractable (FPT) if there is an algorithm for the problem that runs in time f(k)nO(1). The algorithm is said to be a subexponential time parameterized algorithm if f(k) ≤ 2o(k). For an introduction to parameterized algorithms and complexity see the textbooks [10, 11, 22]. In this paper we pose the following question. Does k-Path on planar graphs admit a subexponential time parameterized algorithm which only uses space polynomial in n? We√ give a positive answer to this question by presenting a poly- 2 nomial space, 2O( k log k)nO(1) time algorithm for k-Path restricted to planar graphs. Our algorithm easily generalizes to any family of graphs which exclude a ﬁxed graph H as a minor.

√The fastest parameterized algorithm√ for k-Path on planar graphs runs in 2O( k)nO(1) time and uses 2O( k) log n + nO(1) space [9]. The algorithm we present uses polynomial space, but is slower by a factor of O(log2 k) in the exponent of 2. Is the trade-off worth it? In general, does it make sense to settle for slightly slower algorithms if used space is reduced drastically? We believe that such a trade-off is reasonable because algorithms that use exponential time and space tend to run out of space long before they run out of time. In the survey paper on exponential time algorithms, Woeginger [24] states that “algorithms with exponential space complexities are absolutely useless for real life applications”. This line of reasoning has opened up an interesting research direction; for which problems can we obtain space-efficient algorithms that are (almost) as time-efficient as the fastest ones? Some progress has been made – Fomin, Grandoni and Kratsch [12] gave a 6knO(log k) time, polynomial space algorithm for the Steiner Tree problem and showed how to use it to obtain a O(1.60n) time polynomial space algorithm. In a breakthrough paper, Nederlof [21] gave a 2knO(1) time polynomial space algorithm for Steiner Tree. Subsequently, Lokshtanov and Nederlof [20] devised general sufficient conditions for turning exponential space dynamic programming algorithms into polynomial space algorithms based on algebraic transforms. It is natural to ask which problems admit polynomial space FPT algorithms. While this might look like a whole new research program, the question has a surprising answer; any problem for which a polynomial space FPT algorithm is remotely feasible has one. In particular, a necessary condition for a problem to have a polynomial space FPT algorithm is to have an FPT algorithm. Another necessary condition is that the problem is solvable in polynomial space and time ng(k) for some function g (that is, a polynomial space XP algorithm for the problem). A well-known trick from parameterized complexity shows that these two conditions are not only necessary, but also sufficient. Theorem 1. If a parameterized problem Π has an algorithm A which uses f(k)nc time and space, and an algorithm B which uses ng(k) time and polynomial space, then Π can be solved in nc+1 + f(k)g(k) time and polynomial space. Proof. If n ≥ f(k) then run algorithm A, this takes polynomial time and space. If n ≤ f(k) then run algorithm B, this takes polynomial space and f(k)g(k) time. ut Because of Theorem 1 the right question to ask is not which problems admit polynomial space FPT algorithms, but rather which problems admit fast polynomial space FPT algorithms. Obvious candidates for scrutiny are problems for which the fastest parameterized algorithms require exponential space.

Graph problems restricted to planar and H-minor free graphs have√ this prop- erty – the Bidimensionality theory of Demaine et al. [4] gives 2O( k)nO(1) or √ 2O( k log k)nO(1) time algorithms for a multitude of graph problems on H-minor free graphs. However, these algorithms crucially depend on exponential space dynamic programming algorithms on graphs of bounded treewidth. In particular, the crux of all bidimiensionality based√ algorithms is to ﬁrst bound the treewidth of the input graph by t = O( k), and then solve the problems in 2O(t)n (or tO(t)n) time and 2O(t) log n (or tO(t) log n) space. We refer to following surveys for further details on the the Bidimensionality theory and its several applications [5, 8].

2 Interestingly, using another simple trick we can make most of the algorithms for bidimensional problems on H-minor free graphs run in polynomial space, at the cost of a O(log k) factor in the exponent of the running time. The trick has two components, the first is that (almost) all of the 2O(t)n time and 2O(t log t)n time dynamic programming algorithms on graphs of treewidth t can be turned into polynomial space divide and conquer algorithms with running time nO(t) and nO(t log t). The second component is that most problems for which the Bidi- mensionality theory of Demaine et al. [4] gives fast algorithms admit linear kernels on planar and H-minor-free graphs [3, 14]. A linear kernel for a parameterized graph problem is a polynomial time pre-processing algorithm that takes instances (G, k) and transforms them into equivalent instances (G0, k0) of the same problem such that k0 = O(k) and V (G0) = O(k). Now the subexponential time algorithms of Demaine et al. [4] can be made to run in polynomial space as follows. Run the pre-processing algorithm first, to ensure that the number√ of vertices in the input graph is at most O(k). Bound the treewidth by t = O( k) as before, but replace the 2O(t)n (or tO(t)n) time algorithms by nO(t) (or nO(t log t)) time polynomial space algorithms. Since n = O(k) this second step uses only √ √ 2 2O( k log k) (or 2O( k log k)) time respectively. We remark that for many of these problems one can even remove the log k overhead in the exponent by using a Lipton-Tarjan separator approach [18, 19] instead of the polynomial space algorithms for graphs of bounded treewidth after obtaining the linear kernel for the problem, as done in [1]. In the above argument it was crucial that the problem considered admits a linear kernel. What about the problems that do not? Notably, the k-Path problem does have a subexponential time parameterized algorithm on H-minor-free graphs. At the same time k-Path does not admit a linear (or polynomial size) kernel unless the polynomial hierarchy collapses to the third level [2], even when the input is restricted to planar graphs. Thus, with respect to parameterized polynomial space, subexponential time algorithms for planar graph problems, k-Path stands√ out as a blank spot in an almost chartered map. Our polynomial 2 space, 2O( k log k)nO(1) time algorithm for k-Path on H-minor free graphs fills this gap. Our approach for obtaining polynomial space subexponential time algorithm for k-Path seems applicable to other problems also that do not admit polynomial kernels or not known to admit a polynomial kernel.

2 Deﬁnitions and Notations

In this section we give various deﬁnitions which we make use of in the paper. Let G be a graph then we use V (G) and E(G) to denote its vertex set and the edge set respectively. A graph G0 is a subgraph of G if V (G0) ⊆ V (G) and E(G0) ⊆ E(G). The subgraph G0 is called an induced subgraph of G if E(G0) = {uv ∈ E(G) | u, v ∈ V (G0)}, in this case, G0 is also called the subgraph induced by V 0 and denoted with G[V 0]. By N(u) we denote the (open) neighborhood of u, that is, the set of all vertices adjacent to u. The closed neighbourhood of u is

3 S N[u] = N(u) ∪ {u}. Similarly, for a subset D ⊆ V , we deﬁne N[D] = v∈D N[v] and N(D) = N[D] \ D.

Parameterized algorithms and Treewidth. A parameterized problem Π is a subset of Γ ∗ × N for some ﬁnite alphabet Γ . An instance of a parameterized problem consists of (x, k), where k is called the parameter. A central notion in parameterized complexity is ﬁxed parameter tractability (FPT) which means, for a given instance (x, k), solvability in time f(k) · p(|x|), where f is an arbitrary function of k and p is a polynomial in the input size. A tree decomposition of a graph G is a pair (T, B) where T is a tree and B = {Xi | i ∈ V (T )} is a collection of subsets of V (G) such that S 1. i∈V (T ) Xi = V (G), 2. for each edge xy ∈ E, {x, y} ⊆ Xi for some i ∈ V (T ); 3. for each x ∈ V (G) the set {i | x ∈ Xi} induces a connected subtree of T .

The width of the tree decomposition is maxi∈VT |Xi|−1. The treewidth of a graph G is the minimum width over all tree decompositions of G. A tree decomposition (T, B) can be converted in linear time [17] into a nice tree decomposition of the same width: here, the tree T is rooted and binary, and its nodes are of four types:

– Leaf nodes h are leaves of T and have |Xh| = 1. – Introduce nodes h have one child i with Xh = Xi ∪ {v} for some vertex v ∈ V . – Forget nodes h have one child i with Xh = Xi \{v} for some vertex v ∈ V . – Join nodes h have two children i, j with Xh = Xi = Xj. We denote by tw(G) the treewidth of the graph G.

3 Polynomial Space Algorithm for the k-Path problem

In this section we prove our main result, which is a polynomial space subexponential time algorithm for the k-Path problem. For a set W ⊆ V (G) a set S ⊆ V (G) is a balanced separator for W if V (G) can be partitioned into L, 2|W \S| S and R such that there is no edge from L to R and |W ∩ L| ≤ 3 and 2|W \S| |W ∩ R| ≤ 3 . In other words, W is evenly distributed between L and R. It is well-known that in any tree T , for every set W ⊆ V (G) there is a balanced separator S for W with |S| = 1. This result has been generalized to graphs of bounded treewidth [23][11, Lemma 11.16] - in particular in a graph G of treewidth at most t, for any set W there is a balanced separator S of size at most t + 1. Lemma 1 is a subtle strengthening of this fact and states that given any tree-decomposition of G of width at most t the separator S can be chosen as one of the bags of the decomposition. In fact, the proofs given in [23][11, Lemma 11.16] already imply Lemma 1, we include a proof here for completeness. For a graph G and a nice tree-decomposition (T, B) of G and node v ∈ V (T ) let Xv ∈ B be the corresponding bag. Let Tv be the subtree of T rooted at v and let A(v) = (S X ) \ X . u∈V (Tv ) u v

4 Lemma 1. Let G be a graph, let (T, B) be a nice tree-decomposition of G of width t and let W ⊆ V be a vertex set of size at least 3. Then there exists a vertex v such that Xv is a balanced separator for W and in the corresponding partition V (G) = L ∪ Xv ∪ R, L = A(v). Proof. Recall that T is a rooted tree with root r, and that each node of T |W \Xr | has at most two children. Observe that |A(r) ∩ W | = |W \ Xr| ≥ 3 . |W \Xv | Choose a lowermost node v such that |A(v) ∩ W | ≥ 3 . We prove that 2|W \Xv | 0 |A(v) ∩ W | ≤ 3 . If v is an introduce node or a forget node with child v 0 |W \Xv0 | 2|W \Xv | then |A(v ) ∩ W | ≤ 3 and hence |A(v) ∩ W | ≤ 3 . Here we used that |W | ≥ 3 and that |Xv| = |Xv0 | + 1 if v is an introduce node, and that |A(v)| = |A(v0)| + 1 if v is a forget node. If v is a join node with children u and |W \Xu| w then Xv = Xu = Xw and A(v) = A(u) ∪ A(w). Finally |A(u) ∩ W | ≤ 3 |W \Xw| 2|W \Xv | and |A(w) ∩ W | ≤ 3 and hence |A(v) ∩ W | ≤ 3 . This concludes the proof. ut A key component of our algorithm is a new divide and conquer algorithm for the k-Path problem on graphs of bounded treewidth. Lemma 2. There is an (ntt)O(log k) time and polynomial space algorithm for k-Path if a nice tree-decomposition (T, B) of G of width t is given as input. Proof. We describe a divide and conquer algorithm for the problem. In order to handle the instances that are generated in the recursive steps of the algorithm we will solve a slightly more general problem. In the generalized k-Path problem (k-GP) we are given a graph G, integer k, vertex set Vp and an edge set Ep such that every edge in Ep has both endpoints in Vp. The task is to determine whether there is a path P in G such that Vp ⊆ V (P ), Ep ⊆ E(P ) and |V (P )\Vp| = k. One can think of the sets Vp and Ep as vertices and edges which are pre-determined to be in the path P . We are now ready to give an algorithm for k-GP in graphs 2 t+2 log k+O(1) of bounded tree-width with running time O((nk (t + 1)!3 ) · (Vp + t log k)!). Suppose that there is a path P such that Vp ⊆ V (P ), Ep ⊆ E(P ) and |V (P ) \ Vp| = k. Let W = V (P ) \ Vp, with |W | = k. If k = 0 the algorithm tries all the |Vp|! possible orderings of Vp and checks whether any of the orderings is a path that contains all edges of Ep. If k < 3 the algorithm tries all possible ways to extend Vp by k vertices, and then proceeds to the case when k = 0. We now handle the case that k ≥ 3. By Lemma 1 there exists a node v ∈ V (T ) such that Xv ∈ B is a balanced separator for W . The algorithm guesses the correct vertex v by looping over all the n possible bags in the decomposition. Furthermore, by Lemma 1 there is a partition of V (G) into L ∪ Xv ∪ R such that L = A(v) and R = V (G) \ (Xv ∪ R), 2|W \Xv | 2|W \Xv | there is no edge from L to R, |W ∩ L| ≤ 3 and |W ∩ R| ≤ 3 . The algorithm computes L and R from v. Now the algorithm guesses V (P )∩Xv = X t+1 by trying all the 2 possible subsets of Xv. The path P visits the vertices of X in some order, say x1, x2, . . . xq. The algorithm guesses this order by trying all the (at most) (t + 1)! possible permu- tations of X. Now, for each i < q the subpath of P from xi to xi+1 either uses

5 the edge xixi+1, or has all its inner vertices in L, or has all its inner vertices in R. By trying each of the three possibilities for each i < q the algorithm correctly guesses which of the possibilities it is. The algorithm also guesses whether the subpath of P attached to x1 lies in its entirety in L ∩ X or in R ∩ X. The same guess is performed for the subpath of P attached to xq. Finally the algorithm guesses kL = |(V (P ) \ Vp) ∩ L| and kR = |(V (P ) \ Vp) ∩ R|. L L Using all the guesses the algorithm constructs two instances GL,Vp ,Ep , kL R R L L and GR,Vp ,Ep , L, kR as follows. We set Vp = (Vp ∩ L) ∪ X and Ep = Ep ∩ E(G[L ∪ X]). To construct GL we start with G[L ∪ X] and remove all the edges with both endpoints in X. For every pair xi, xi+1 such that we have guessed that the subpath of P from xi to xi+1 uses the edge xixi+1 or has all its internal L vertices in R, we add the edge xixi+1 to GL and to Ep . Finally, if we have guessed that the subpath of P from the start point until x1 lies entirely in X ∪R L we add a vertex pstart to GL and to Vp , make pstart adjacent to x1 and add L p1pstart to Ep . Similarly if we have guessed that the subpath of P from xq to the end point of P lies entirely in X ∪ R we add a vertex pend to GL and to L L R Vp , and add the edge xqpend to GL and to Ep . The graph GR and set Vp is constructed symmetrically. L L R R If the two instances GL,Vp ,Ep , kL and GR,Vp ,Ep , kR are both “yes” instances one can glue their solution paths together to form a solution path for G, Vp,Ep, k. In particular every edge xixi+1 in GL will correspond either to an L L R R edge xixi+1 in the solution paths of both GL,Vp ,Ep , kL and GR,Vp ,Ep , kR, R R or it can be replaced by a subpath of the solution path of GR,Vp ,Ep , kR. Edges xixi+1 in GR are handled symmetrically. In the reverse direction for the correct L L set of guesses the path P breaks up into solution paths to GL,Vp ,Ep , kL and R R GR,Vp ,Ep , kR respectively. On the side that contains pstart we add the edge pstartx1 to the solution path, and on the side that contains pend we add the edge xqpend. Now we bound the running time of the algorithm. Observe that since we only add edges between vertices in X ⊆ Xv and pendant vertices pstart and pend of degree 1 attached to X, the treewidth of GL and GR is at most t. Let r = |Vp|, and let T (k, r, n, t) be a function that upper bounds the running time of the algorithm. The function T is bounded by the following recurrence.

T (k, r, n, t) ≤ n · (t + 1)! · 3t · 2t+3 · k2 · 2T (k/2, r + t, n, t) when k ≥ 3

Here the factors in the recurrence reﬂect the number of possibilities for each guess, except for the factor 2 in 2T (k/2, n, t) which reﬂects that GL and GR are handled independently. Observe that n and t never increase throughout the recurrence. When k < 3 we have that T (k, r, n, t) ≤ O(n3r!). Hence T (k, r, n, t) can be bounded from above by

T (k, r, n, t) ≤ (nk2(t + 1)!6t+3)log k · n3(t log k)! ≤ (ntt)O(log k).

The space requirement of the algorithm is clearly polynomial. This concludes the proof. ut

6 We are now in position to give the main result of this section.

Theorem 2. For every ﬁxed graph H√, there is an algorithm for k-Path on H- 2 minor-free graphs running in time 2O( k log k)nO(1) and using polynomial space.

Proof. We use the fact that for any H there exists a constant h such that in any H-minor-free graph√ of treewidth at least hk there is a k-path of length at least k2 [4]. Set t = h k, then if tw(G) ≥ t then G contains a k-path. We use the approximation algorithm of Diestel et al [6] to either compute a tree- decomposition of G of width at most 3t/2, or to conclude that the treewidth of G is at least t, which implies that G has a k-path. Diestel et al’s algorithm O(t) O(1) uses polynomial space and 2 √n time. If we obtain a tree-decomposition k we proceed as follows. If √n ≤ 2 apply the algorithm from Lemma 2 to solve O( k log2 k) O(1) the problem in time√ 2 n and using polynomial space. If, on the k other hand n ≥ 2 the standard dynamic√ programming algorithm on graphs of bounded treewidth that uses at most 2O( k)nO(1) time and space [7], runs in polynomial time and space. This concludes the proof. ut

We remark that the√ algorithm presented in Theorem 2 for k-Path can be made to run in time 2O( k log k)nO(1) and space polynomial in n on planar graphs using sphere cut decompositions introduced by Dorn et al. [9].

4 Conclusion and Discussion

We gave a subexponential time, polynomial space parameterized algorithm for the k-Path problem on H-minor-free graphs. A key component of our algorithm is a new (n·tt)O(log k) time and polynomial space algorithm for k-Path in graphs of treewidth at most t. In general, it is possible to design similar (n·2t)O(log k) or (n·tt)O(log k) time and polynomial space algorithms for many problems where one is looking for a speciﬁc vertex set of size k in a graph of treewidth t. A concrete example where this is useful is the k-Partial Vertex Cover problem. Here we are given a graph G, positive integers k and t and we look for a subset S ⊆ V (G) such that |S| ≤ k and the number of edges incident to S is at least t. This problem is not known to admit a polynomial kernel, even on planar graphs. However using the approach described in this paper for k-Path and combining it with an algorithm of Fomin et al. [13] that in polynomial time either ﬁnds 0 a solution for an instance√ (G, k, t) or obtains an equivalent instance (G , k, t) such that tw(G0) ≤ O( k), one can give a subexponential time, polynomial space parameterized algorithm for k-Partial Vertex Cover on apex-minor- free graphs. We conclude with two open problems. First, is there a polynomial space parameterized√ algorithm for the k-Path problem on planar graphs with running time 2O( k)nO(1)? Second, by combining the well known 2tnO(1) time and space algorithm for Independent Set in graphs of treewidth t, and the folklore nO(t) 2 time, polynomial space algorithm, Theorem 1 yields a 2O(t ) + n2 time and

7 polynomial space algorithm for Independent Set. Is there a polynomial space algorithm for Independent Set on graphs of treewidth t with running time 2− 2t nO(1) for some > 0?.

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